A Java Servlet container can be multithreaded and to allow servlets to run across containers it needs to be thread safe but there is nothing that saids it has to be multi-threaded per se. Anyway all well known are.
Anyway J2EE is death
we started calling newer versions JEE ages ago.
So to actually be constructive, if I had to go about it I would forget about the 3 tier cause it makes no sense for short-lived stuff like a poker system apart from winnings and other stuff you need to track but you might want to focus on getting things to work first. Given this I would look at the channel api and async responses introduced in JEE6 you can go ajax or html5 canvas easily. Hell have a look at vaadin while your at it.